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Well, that depends. (And yes I get the joke.) do we actually talk about probability, likelihood, or both, and shall we treat 'everything' (not really everything, but I feel too lazy to type this out now) as in the sample space, in the parameter space, or does it depend?

Consider an infinite plane covered with parallel lines spaced 1 unit apart. At a random position on this plane, drop a 1-unit-long needle at a random angle. What is the probability the needle lies across one of the lines?

I know very little math, but I suspect it's because e is representative for compounding stuff, so when you have a problem that can be reduced to something compounding your answer is somehow related to e.

This idea is about as solid as an alien conspiracy theory, in other words just a random guess.

There's a really clever method to show that the probability of exceeding 1 after k trials is 1 - 1/k!, which doesn't require calculus or geometry.

Consider the k partial sums. That is, let S(j) be the sum of the first j random numbers, for 0 < j <= k. We want to show that the probability that S(k) < 1 is 1/k!.

S(j+1) is uniformly distributed in the interval [S(j), S(j)+1]. Consider the fractional part F(j) = S(j) - floor(S(j)). Because it wraps around, F(j+1) is uniformly distributed in the interval [0, 1), regardless of the value of F(j).

So we've got k values of F(j) uniformly distributed in [0, 1). Finally notice that S(k) < 1 if and only if the F(j)'s are sorted in increasing order. The probability that k uniform random values in [0, 1) will be chosen in increasing order is 1/k!.

I actually tackled this problem for fun, during a course called "Computer Intensive Statistical Methods". Looking at the proofs you provided, I'd say that my method was overkill. The proof is in PDF format below:

EDIT: Removed the link with the popups. Good reminder to always try to download before linking.